Question

Assertion :$\alpha ,\beta$ are the roots of the equation $x^2-7x+2=0,r$ is the arithmetic mean of $\alpha ,\beta$ The locus of the point $P$ which is equidistant from the points $(\alpha ,\beta )$ and $(\alpha +r,\beta +r)$ is $2x+2y-21=0.$ Reason: Locus of a point which moves such that the sum of its distances from two fixed is constant always passes through the mid-point of the line joining the fixed points.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Assertion:
$\alpha +\beta =7,r=\frac{7}{2}$
Locus is
${(x-\alpha )}^2+{(y-\beta )}^2={(x-(\alpha +r))}^2+{(y-(\beta +r))}^2\Rightarrow x+y-(\alpha +\beta )=r\Rightarrow 2x+2y-21=0$
Hence it is true.
Reason:
Let the fixed point be $(-1,0),(1,0)$ whose mid point is $(0,0)$
Locus is
$\sqrt{{(x+1)}^2+y^2}+\sqrt{{(x-1)}^2+y^2}=c$
which does not pass through $(0,0)$ for all value of c.
Hence reason is false.